GCSE Maths / Edexcel

3D Pythagoras and trigonometry

Use hidden right-angled triangles in 3D shapes to find lengths and angles.

Geometry and MeasuresHigherGrades 6 to 9Skill

Curriculum path: GCSE Maths > Edexcel > Geometry and Measures > 3D Pythagoras and trigonometry

Pearson Edexcel GCSE Maths Higher geometry G20 and G22: apply Pythagoras' theorem and trigonometry in 3D contexts.

Revision notes

Theory, examples, and quick checks.

Keep the method short, then practise straight away. This note is written for GCSE Maths Edexcel students who need clear working and reliable method marks.

Theory

3D Pythagoras and 3D trigonometry questions usually become two ordinary right-angled triangle questions.

Do not try to use every length in the 3D shape at once. First find the right triangle that sits on a face or across the base.

In a cuboid, the base diagonal is often found first using Pythagoras on the base rectangle.

After finding the base diagonal, combine it with the height to make a second right-angled triangle through the solid.

For an angle of elevation in a cuboid, the horizontal distance is often the base diagonal, not just one edge.

Higher note: draw the hidden triangle separately if the 3D diagram feels crowded.

Edexcel questions often award marks for the intermediate length, so show the base diagonal before the final answer.

Key ruleFind the hidden 2D right triangle first. In cuboids, base diagonal first, then space diagonal or angle.

Diagram guide

base diagonal firstspace diagonalθ3D trig is usually two right trianglesfind a hidden length before using trig
Hidden right trianglesThe base diagonal creates the second right triangle needed for the space diagonal or angle.
AG8 cm6 cm5 cmFind the base diagonal first, then the space diagonal.
3D Pythagoras linkUse Pythagoras twice, or use the direct cuboid diagonal relationship after you understand where it comes from.

Worked examples

Find a space diagonal

A cuboid is 8 cm by 6 cm by 5 cm. Find the space diagonal.

base diagonal firstspace diagonalθ3D trig is usually two right trianglesfind a hidden length before using trig
Example: cuboid diagonalFind the base diagonal first, then use it with the height.
  1. Base diagonal² = 8² + 6² = 100.
  2. Base diagonal = 10 cm.
  3. Space diagonal² = 10² + 5² = 125.
  4. Space diagonal = √125 = 11.2 cm to 3 significant figures.

Answer: 11.2 cm

Find an angle

A cuboid is 12 cm by 5 cm by 9 cm. Find the angle the space diagonal makes with the base.

base diagonal firstspace diagonalθ3D trig is usually two right trianglesfind a hidden length before using trig
Example: angle with the baseThe adjacent side is the base diagonal.
  1. Base diagonal² = 12² + 5² = 169, so base diagonal = 13 cm.
  2. Use tan θ = opposite / adjacent = 9 / 13.
  3. θ = tan⁻¹(9 / 13).
  4. θ = 34.7 degrees to 3 significant figures.

Answer: 34.7 degrees

Use a face diagonal

A square-based pyramid has vertical height 10 cm and base side 8 cm. Find the slant height from the centre of one base side to the apex.

base diagonal firstspace diagonalθ3D trig is usually two right trianglesfind a hidden length before using trig
Example: pyramid cross-sectionUse the right triangle through the centre of the base.
  1. From the centre of the square to the centre of a side is 4 cm.
  2. The vertical height is 10 cm.
  3. Slant height² = 10² + 4² = 116.
  4. Slant height = √116 = 10.8 cm to 3 significant figures.

Answer: 10.8 cm

Common mistakes

  • Using one edge as the horizontal distance when the base diagonal is needed.
  • Trying to use SOH CAH TOA before finding the missing side in the hidden triangle.
  • Mixing up face diagonals and space diagonals.
  • Rounding the base diagonal too early.
  • Not drawing the 2D right triangle separately.

Quick exercise

Try these before moving to the exam-style questions.

  1. A cuboid base is 3 cm by 4 cm. What is the base diagonal?
    base diagonal firstspace diagonalθ3D trig is usually two right trianglesfind a hidden length before using trig
    Quick check: base diagonalUse Pythagoras on the base rectangle.
  2. A cuboid has base diagonal 10 cm and height 24 cm. Find the space diagonal.
    AG8 cm6 cm5 cmFind the base diagonal first, then the space diagonal.
    Quick check: second triangleNow use the base diagonal with the height.
  3. For an angle with the base, which length is usually adjacent: height or base diagonal?
    base diagonal firstspace diagonalθ3D trig is usually two right trianglesfind a hidden length before using trig
    Quick check: angle setupThe base diagonal lies along the base.
  4. A cuboid has base diagonal 12 cm and height 5 cm. Find the angle the space diagonal makes with the base.
    base diagonal firstspace diagonalθ3D trig is usually two right trianglesfind a hidden length before using trig
    Quick check: tangent angleUse tan θ = opposite / adjacent.
  5. Why should you avoid rounding the base diagonal too early?
    base diagonal firstspace diagonalθ3D trig is usually two right trianglesfind a hidden length before using trig
    Quick check: accuracyEarly rounding can damage the final answer.
Exam-style questions

Practise the same skill at three levels.

These are original GCSE-style questions with mark schemes, common wrong answers, and AI marking guidance so feedback stays close to exam expectations.

Basic GCSE styleHigherCalculator4 marks

A cuboid measures 9 cm by 12 cm by 8 cm. Find the length of its space diagonal.

base diagonal firstspace diagonalθ3D trig is usually two right trianglesfind a hidden length before using trig
Question diagram: space diagonalUse the base diagonal first.
3D Pythagorascuboidhigher geometry
Standard exam styleHigherCalculator5 marks

A cuboid is 7 cm by 24 cm by 10 cm. Work out the angle the space diagonal makes with the base, to 1 decimal place.

base diagonal firstspace diagonalθ3D trig is usually two right trianglesfind a hidden length before using trig
Question diagram: angle with baseThe adjacent side is the base diagonal.
3D trigonometryangle of elevationcuboid
ChallengeHigherCalculator5 marks

A square-based pyramid has base side 14 cm and vertical height 18 cm. Find the angle between a sloping edge from a corner to the apex and the base.

base diagonal firstspace diagonalθ3D trig is usually two right trianglesfind a hidden length before using trig
Question diagram: pyramid angleUse the distance from the centre of the base to a corner as the adjacent side.
3D trigonometrypyramidhigher problem solving